On the topology of 3-manifolds admitting Morse-Smale diffeomorphisms with four fixed points of pairwise different Morse indices

TL;DR

This paper studies the topology of 3-manifolds supporting Morse-Smale diffeomorphisms with four fixed points of distinct Morse indices, revealing conditions for lens space structures and constructing examples with wild embeddings.

Contribution

It extends previous results by showing that all such diffeomorphisms support lens spaces and constructs new examples with wild embeddings on any lens space.

Findings

Supporting manifold is a lens space $L_{p,q}$ for tame embeddings.

Existence of diffeomorphisms with wild embedded separatrices on all lens spaces.

Wandering set contains at least $p$ non-compact heteroclinic curves.

Abstract

In the present paper we consider class $G$ of orientation preserving Morse-Smale diffeomorphisms $f$, which defined on closed 3-manifold $M^3$, and whose non-wandering set consist of four fixed points with pairwise different Morse indices. It follows from S. Smale and K. Meyer results that all gradient-like flows with similar properties has Morse energy function with four critical points of pairwise different Morse indices. This implies, that supporting manifold $M^3$ for these flows admits a Heegaard decomposition of genus 1 and hence it is homeomorphic to a lens space $L_{p,q}$. Despite the simple structure of the non-wandering set in class $G$ there exist diffeomorphisms with wild embedded separatrices. According to V. Grines, F. Laudenbach, O. Pochinka results such diffeomorphisms do not possesses an energy function, and question about topology their supporting manifold is open. According to V. Grines, E. Zhuzhoma and V. Medvedev results $M^3$ is homeomorphic to a lens space $L_{p,q}$ in case of tame embedding of closures of one-dimensional separatrices of diffeomorphism $f\in G$. Moreover, the wandering set of $f$ contains at least $p$ non-compact heteroclinic curves. In the present paper similar result was received for arbitrary diffeomorphisms of class $G$. Also we construct diffeomorphisms from $G$ with wild embedding one-dimensional separatrices on every lens space $L_{p,q}$. Such examples were known previously only on the 3-sphere.